I am reading a proof that a monotonic function in closed interval is integrable.
The proof uses the assumption that "Let assume that $f$ is increasing, therefore for all $x\in[a,b]$: $f(a)\le f(x)\le f(b)$ and $f$ is bounded"
Why can we assume that $f$ is bounded? maybe $\lim_{x\to b}f(x)=\infty$?
Moreover if every monotonic function in closed interval is bounded then it also get it maximum and minimum values there?
Best Answer
Because the function is defined on a closed interval. Since it is increasing, certainly its value at $b$ is maximum.
If we had that $\lim_{x\to b}=+\infty$, then either the function is not defined at $b$ (so the interval is not closed), or whichever (finite) value one assigns to $f$ at $b$ will violate monotonicty.